The task design model used in Paper 2

Paper 2 provides a task design model of so-called prediction tasks intended to foster student reasoning concerning exponential func- tions in a DMS environment. The prediction tasks are embedded in a task sequence with the aim of developing students’ awareness of the difference between linear and exponential growth. The literature pro- vided a framework for the initial task design, which comprises ideas of how to encourage students to perform different aspects of reasoning in relation to functions and graphs. This section follows the same structure as the previous one; first, the guiding principles are intro- duced, followed by a description of the initial version of the task de- sign model and finally, the suggested refinements are discussed.

The guiding principles

In the design of prediction tasks, in total seven design principles guid- ed by the literature (Section 3.2) were used. These are logical flow, unexpected outcomes, mathematical reality, fostering instrumental genesis, formulations in writing, and working in pairs.

Logical flow. The structuring of the prediction task should promote a logical flow by first requesting students to make a prediction about an outcome of a mathematical situation, and then to investigate the situ- ation under consideration. Finally, the students are encouraged to re- flect on and/or explain the outcome of the investigation in relation to their initial prediction.

Unexpected outcomes. Particularly suitable as prediction tasks are those capable of eliciting some misconception common among stu- dents or those that are considered challenging for many students, and hence might provide unexpected outcomes.

Mathematical reality. The tasks in the task sequence, in which the prediction tasks are embedded, should be related to a common theme, preferably connected to a context that students experience as rea- listic. The reason for this is to enhance the opportunity for students to make reflections and connections between different tasks in a task

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sequence and to offer students a familiar situation and thereby in- crease their motivation.

Fostering instrumental genesis. The tasks were to be intertwined with computer instructions since students were expected to do the con- structions by themselves, that is, they should not use prepared applets. The main reason for this choice is to enhance the scope for the mathematics software to become an instrument for the students.

Formulations in writing. Students were, when appropriate, expected to formulate their predictions and explanations in writing to make them reflect further on the situation they are working on. Moreover, in this way their reasoning becomes public to others to be used in a follow-up whole-class discussion.

Working in pairs. Students were encouraged to work in pairs at one computer. In this way, the results from explorations with the software are displayed on a common computer screen, which serves as a com- mon referent to enhance joint reasoning. This was important because one main purpose with the prediction tasks was to foster student rea- soning.

Didactical variables. The domain-specific literature related to func- tions and graphs in a dynamic graphical software environment (see Section 5.3), served as guidance in the a priori identification of four didactical variables. While tackling the tasks, students might (a) use different forms of representation of functions and make translations between them, (b) experience both a local and a global view of func- tions, (c) use both a correspondence and covariation approach to functions and (d) pay attention to the effects of scaling of coordinate axes.

The initial version

In total three different prediction tasks, embedded in a task sequence, were trialled with four tenth grade classes, involving a total of 85 stu- dents. To articulate the theoretical rationale for different design choices that might affect students’ reasoning and to analyse them

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after empirical testing, the design tool of didactical variables was em- ployed. In total, seven didactical variables based on the literature were identified a priori, and during the analysis process four further didactical variables were identified. Besides the four domain-specific didactical variables mentioned above, one variable concerned which tools, paper and pencil or computer students were asked to use to solve a particular task. Still another variable concerned whether to ask students for explanations or not.

Moreover, the conjectured local instruction theory that goes with the designed tasks comprises a detailed description about expected stu- dent reasoning which might occur throughout the implementation of the activities in the classroom.

Suggested refinements

Besides providing suggestions for revision of the particular prediction tasks, Paper 2 pinpoints key didactical variables that proved im- portant to consider in designing prediction tasks. These didactical variables were grouped into the following three concerns: explana- tions, functions and scaffolding issues. Below follows a description of how some of the didactical variables provided guidance in the refine- ment of the prediction tasks to enhance the opportunity for students to develop reasoning based on conceptual understanding.

The didactical variables concerning explanations turned out to be cru- cial in the design of prediction tasks. The a priori didactical variable identified Ask for an explanation or not was discussed in all predic- tion tasks, and in the revised version we suggest adding some further requests for explanations. The choice whether to ask for an explana- tion or not was essential both in the prediction part and the compari- son part of each prediction task. However, sometimes it might be ap- propriate to just ask for a description or a comment. Further, the re- sults indicate that the wording is crucial in the formulation of ques- tions where students are asked for explanations. This implied the identification of a new didactical variable regarding how to direct stu- dents’ focus on what to explain. Actually, all requests for explanations in the study turned out to require some refinement concerning this

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didactical variable. In the revised versions, we endeavour to make the requests more pointed.

The didactical variable concerning what form of representation to use is elaborated on in all prediction tasks. For instance, if the aim is to provoke a known misconception, it is important to reflect on whether this misconception interrelates to a specific form of representation. The didactical variable concerning the choice of covariation or corre- spondence approach turned out to be essential. In several cases revi- sions were made to emphasize a covariation approach. The main rea- son for these revisions was to encourage students to reflect on the mathematics behind exponential growth and thereby promote con- ceptual reasoning.

Scaffolding issues were elaborated on in all prediction tasks and all the new didactical variables identified during the design process are more or less related to these issues. For example, one didactical varia- bleconcerns the question whether to support students by specifying the scaling of axes or not. Another didactical variable was introduced in the revision of one of the tasks where students were given a numer- ic value determining what to count as a good enough guess. Thus, whether to give this scaffolding or not became a new didactical varia- ble.